The formation of Galaxy Clusters from density fluctuations in the early Universe

May I ask for a little assistance in filling in the gaps of my understanding? :S
I understand that the current large scale structure of the Universe is thought to be the result of early fluctuations in density which have been stretched out in the expansion of the Universe.

I am a bit muddled as to why you can model these density fluctuations (CDM model) as Gaussian at 'decoupling'.

1. If decoupling is the moment in recombination at which Compton scattering became less than the expansion of the universe, how does that make it Gaussian? Is it just based on the fact that the photons stopped interactions with matter at this point, such that there would be a spread of velocities?

The Wik. is often very helpful, it is a wonderful resource, but you need to be careful. Sometimes it is written in an unclear way and can even be confusing or have misinformation.

This LBL.gov link is to some public outreach by the "Smoot Group" at Lawrence Berkeley Lab. George Smoot is a Nobel laureate cosmologist. He is a worldclass expert on structure formation in early U.
That doesn't mean that he or his group are RIGHT, but their stuff is probably a bit more reliable than the Wik.

... If decoupling is the moment in recombination at which Compton scattering became less than the expansion of the universe, how does ...?

It's not clear to me what that even means! And it is straight out of Wik, so any confusion of concepts or language is not your fault.

If we go by the LBL Smoot Group webpage, then "decoupling" means the moment of "decoupling of light and matter", when the partially ionized (mostly hydrogen) gas cooled down to around 3000 kelvin and became effectively transparent.

There are other people who call that same moment "recombination"!

In other words when Wikipedia says "...decoupling is the moment in recombination at which Compton scattering became less than the expansion of the universe.." they are confusing people. It is not generally accepted language to say that decoupling is a "moment in" some larger process called recomb.
For a lot of people the two terms are just synonyms.

And the physical description is confusing. What does it mean to say "Compton scattering became less than the expansion"??
=====================

So there are two needs here, two areas of question, that should have response.

What actually happened in this partly ionized gas to make it turn effectively transparent when it had expanded and cooled to 3000 kelvin? And why wasn't transparent earlier when it was hotter?

How come very slight random fluctuations of density and temperature were left over after the supposed period of inflation. And how come some 380,000 years later they were still there (when she turned transparent) and of all different sizes?

Is that a fair summary of what you are wondering about? Several people here might respond but I just want first to be clear about the question.

BTW Smoot has a great 18 minute video talk (for general audience) about structure formation from density fluctuations. Wisps of dark matter gathered before even galaxy clusters. Check it out, some good visuals. Google "Smoot TED"
It was a talk give to an organization called TED.

Would the regions of overdensity already have been established before this "moment of decoupling of light and matter", and now since it has become a gas(??) you can treat it as Gaussian?

Yes, and this is a very important point to emphasize. It helps to realized that the initial perturbations were generated during inflation, at which time they were in the form of perturbations in the curvature of spacetime (there is no matter or radiation to speak of at this point). Then, after inflation, the universe reheats -- it becomes repopulated with matter and radiation -- and our standard picture of Big Bang cosmology picks up from this point. Now, this matter and radiation comprised ordinary, or "baryonic" matter, photons, and dark matter (among other things not germane to this discussion). Now, the dark matter behaves very differently from the ordinary matter, in that it does not couple to the photons. So, as soon as the universe finishes reheating, the dark matter begins to 'feel' the curvature perturbations, and begins to clump in the gravitational potential wells they create. Of course, the ordinary matter also feels these perturbations, but they are too tightly bound to the photons to fall in. Instead, they oscillate in and out in a cosmic acoustic wave -- gravity on the one hand pulling them in, photon pressure pushing them out, all the while, the dark matter is just clumping away.

When the ordinary matter and photons finally decouple, the ordinary matter is free to do its business, and it too begins to clump. If the dark matter didn't get that head start to forming structure -- if structures only started forming when the baryons finally had a chance to join in -- then today's cosmological structures would not be as developed as we see them to be. And so the overdensities had been established by the time of decoupling, and this is one very important piece of evidence for dark matter.

Lastly, about the Gaussianity of the perturbations. This has to do with how inflation generates the perturbations in the first place. This is a fascinating story, and unfortunately, I can't do justice to it here. The curvature perturbations originated as tiny quantum vacuum fluctuations. Inflation acts as a powerful microscope -- it stretches everything out to colossal scales. So, these tiny fluctuations get stretched out by inflation, and they manifest themselves as curvature perturbations. Now, I've just glossed over hundreds of pages of sweat and tears, but that's the gist of it. Now, if you recall from QM, free field vacuum fluctuations are Gaussian. So, the statistical nature of the quantum vacuum gets transferred to the density perturbations -- they retain this Gaussianity. Of course, once these perturbations start evolving and interacting with things, this Gaussianity can (and often does) get spoiled. There are also more complicated inflation models that generate non-Gaussian fluctuations from the get-go. Hope this helps!